Found problems: 260
2009 AMC 12/AHSME, 11
On Monday, Millie puts a quart of seeds, $ 25\%$ of which are millet, into a bird feeder. On each successive day she adds another quart of the same mix of seeds without removing any seeds that are left. Each day the birds eat only $ 25\%$ of the millet in the feeder, but they eat all of the other seeds. On which day, just after Millie has placed the seeds, will the birds find that more than half the seeds in the feeder are millet?
$ \textbf{(A)}\ \text{Tuesday}\qquad \textbf{(B)}\ \text{Wednesday}\qquad \textbf{(C)}\ \text{Thursday} \qquad \textbf{(D)}\ \text{Friday}\qquad \textbf{(E)}\ \text{Saturday}$
2015 AMC 8, 11
In the small country of Mathland, all automobile license plates have four symbols. The first must be a vowel (A, E, I, O, or U), the second and third must be two different letters among the 21 non-vowels, and the fourth must be a digit (0 through 9). If the symbols are chosen at random subject to these conditions, what is the probability that the plate will read "AMC8"?
$
\textbf{(A) } \frac{1}{22,050} \qquad
\textbf{(B) } \frac{1}{21,000}\qquad
\textbf{(C) } \frac{1}{10,500}\qquad
\textbf{(D) } \frac{1}{2,100} \qquad
\textbf{(E) } \frac{1}{1,050}
$
2020 AMC 8 -, 22
When a positive integer $N$ is fed into a machine, the output is a number calculated according to the rule shown below.
[asy] size(300); defaultpen(linewidth(0.8)+fontsize(13)); real r = 0.05; draw((0.9,0)--(3.5,0),EndArrow(size=7)); filldraw((4,2.5)--(7,2.5)--(7,-2.5)--(4,-2.5)--cycle,gray(0.65)); fill(circle((5.5,1.25),0.8),white); fill(circle((5.5,1.25),0.5),gray(0.65)); fill((4.3,-r)--(6.7,-r)--(6.7,-1-r)--(4.3,-1-r)--cycle,white); fill((4.3,-1.25+r)--(6.7,-1.25+r)--(6.7,-2.25+r)--(4.3,-2.25+r)--cycle,white); fill((4.6,-0.25-r)--(6.4,-0.25-r)--(6.4,-0.75-r)--(4.6,-0.75-r)--cycle,gray(0.65)); fill((4.6,-1.5+r)--(6.4,-1.5+r)--(6.4,-2+r)--(4.6,-2+r)--cycle,gray(0.65)); label("$N$",(0.45,0)); draw((7.5,1.25)--(11.25,1.25),EndArrow(size=7)); draw((7.5,-1.25)--(11.25,-1.25),EndArrow(size=7)); label("if $N$ is even",(9.25,1.25),N); label("if $N$ is odd",(9.25,-1.25),N); label("$\frac N2$",(12,1.25)); label("$3N+1$",(12.6,-1.25)); [/asy]
For example, starting with an input of $N = 7$, the machine will output $3 \cdot 7 + 1 = 22$. Then if the output is repeatedly inserted into the machine five more times, the final output is $26$. $$ 7 \to 22 \to 11 \to 34 \to 17 \to 52 \to 26$$ When the same 6-step process is applied to a different starting value of $N$, the final output is $1$. What is the sum of all such integers $N$? $$ N \to \_\_ \to \_\_ \to \_\_ \to \_\_ \to \_\_ \to 1$$
$\textbf{(A)}\ 73 \qquad \textbf{(B)}\ 74 \qquad \textbf{(C)}\ 75 \qquad \textbf{(D)}\ 82 \qquad \textbf{(E)}\ 83$
2016 AMC 8, 2
In rectangle $ABCD$, $AB=6$ and $AD=8$. Point $M$ is the midpoint of $\overline{AD}$. What is the area of $\triangle AMC$?
$\textbf{(A) }12\qquad\textbf{(B) }15\qquad\textbf{(C) }18\qquad\textbf{(D) }20\qquad \textbf{(E) }24$
2017 AMC 8, 11
A square-shaped floor is covered with congruent square tiles. If the total number of tiles that lie on the two diagonals is 37, how many tiles cover the floor?
$\textbf{(A) }148\qquad\textbf{(B) }324\qquad\textbf{(C) }361\qquad\textbf{(D) }1296\qquad\textbf{(E) }1369$
2017 AMC 8, 3
What is the value of the expression $\sqrt{16\sqrt{8\sqrt{4}}}$?
$\textbf{(A) }4\qquad\textbf{(B) }4\sqrt{2}\qquad\textbf{(C) }8\qquad\textbf{(D) }8\sqrt{2}\qquad\textbf{(E) }16$
2023 AMC 8, 22
In a sequence of positive integers, each term after the second is the product of the previous two terms. The sixth term in the sequence is 4000. What is the first term?
$\textbf{(A) }1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 4 \qquad \textbf{(D) } 5 \qquad \textbf{(E) } 10$
2016 AMC 8, 3
Four students take an exam. Three of their scores are $70, 80,$ and $90$. If the average of their four scores is $70$, then what is the remaining score?
$\textbf{(A) }40\qquad\textbf{(B) }50\qquad\textbf{(C) }55\qquad\textbf{(D) }60\qquad \textbf{(E) }70$
2022 AMC 8 -, 6
Three positive integers are equally spaced on a number line. The middle number is $15$, and the largest number is $4$ times the smallest number. What is the smallest of these three numbers?
$\textbf{(A)} ~4\qquad\textbf{(B)} ~5\qquad\textbf{(C)} ~6\qquad\textbf{(D)} ~7\qquad\textbf{(E)} ~8\qquad$
2020 AMC 8 -, 4
Three hexagons of increasing size are shown below. Suppose the dot pattern continues so that each successive hexagon contains one more band of dots. How many dots are in the next hexagon?
[asy]
// diagram by SirCalcsALot
size(250);
real side1 = 1.5;
real side2 = 4.0;
real side3 = 6.5;
real pos = 2.5;
pair s1 = (-10,-2.19);
pair s2 = (15,2.19);
pen grey1 = rgb(100/256, 100/256, 100/256);
pen grey2 = rgb(183/256, 183/256, 183/256);
fill(circle(origin + s1, 1), grey1);
for (int i = 0; i < 6; ++i) {
draw(side1*dir(60*i)+s1--side1*dir(60*i-60)+s1,linewidth(1.25));
}
fill(circle(origin, 1), grey1);
for (int i = 0; i < 6; ++i) {
fill(circle(pos*dir(60*i),1), grey2);
draw(side2*dir(60*i)--side2*dir(60*i-60),linewidth(1.25));
}
fill(circle(origin+s2, 1), grey1);
for (int i = 0; i < 6; ++i) {
fill(circle(pos*dir(60*i)+s2,1), grey2);
fill(circle(2*pos*dir(60*i)+s2,1), grey1);
fill(circle(sqrt(3)*pos*dir(60*i+30)+s2,1), grey1);
draw(side3*dir(60*i)+s2--side3*dir(60*i-60)+s2,linewidth(1.25));
}
[/asy]
$\textbf{(A)}\ 35 \qquad \textbf{(B)}\ 37 \qquad \textbf{(C)}\ 39 \qquad \textbf{(D)}\ 43 \qquad \textbf{(E)}\ 49$
2022 AMC 8 -, 2
Consider these two operations:
\begin{align*}
a \, \blacklozenge \, b &= a^2 - b^2\\
a \, \bigstar \, b &= (a - b)^2
\end{align*}
What is the value of $(5 \, \blacklozenge \, 3) \, \bigstar \, 6?$
$\textbf{(A) } {-}20\qquad\textbf{(B) } 4\qquad\textbf{(C) } 16\qquad\textbf{(D) } 100\qquad\textbf{(E) } 220$
2023 AMC 8, 15
Viswam walks half a mile to get to school each day. His route consists of $10$ city blocks of equal length and he takes one minute to walk each block. Today, after walking $5$ blocks, Viswam discovers he has to take a detour, walking $3$ blocks of equal length instead of one block to reach the next corner. From the time he starts his detour what speed, in mph, must he walk in order to get to school at his usual time?
$\textbf{(A)}~4.0\qquad\textbf{(B)}~4.2\qquad\textbf{(C)}~4.5\qquad\textbf{(D)}~4.8\qquad\textbf{(E)}~5.0\qquad$
2019 AMC 8, 10
The diagram shows the number of students at soccer practice each weekday during last week. After computing the mean and median values, Coach discovers that there were actually $21$ participants on Wednesday. Which of the following statements describes the change in the mean and median after the correction is made?
[asy]
unitsize(1 cm);
real unitwidth, dayheight, barheight;
int i;
unitwidth = 0.5;
dayheight = 1;
barheight = 0.3;
draw((unitwidth,0)--(unitwidth,5*dayheight),gray(0.7));
draw((2*unitwidth,0)--(2*unitwidth,5*dayheight),gray(0.7));
draw((3*unitwidth,0)--(3*unitwidth,5*dayheight),gray(0.7));
draw((4*unitwidth,0)--(4*unitwidth,5*dayheight),gray(0.7));
draw((5*unitwidth,0)--(5*unitwidth,5*dayheight),gray(0.7));
draw((6*unitwidth,0)--(6*unitwidth,5*dayheight),gray(0.7));
draw((7*unitwidth,0)--(7*unitwidth,5*dayheight),gray(0.7));
fill((0,1/2*dayheight - 1/2*barheight)--(8*unitwidth,1/2*dayheight - 1/2*barheight)--(8*unitwidth,1/2*dayheight + 1/2*barheight)--(0,1/2*dayheight + 1/2*barheight)--cycle,gray(0.5));
fill((0,5/2*dayheight - 1/2*barheight)--(8*unitwidth,5/2*dayheight - 1/2*barheight)--(8*unitwidth,5/2*dayheight + 1/2*barheight)--(0,5/2*dayheight + 1/2*barheight)--cycle,gray(0.5));
draw((8*unitwidth,0)--(8*unitwidth,5*dayheight),gray(0.7));
draw((9*unitwidth,0)--(9*unitwidth,5*dayheight),gray(0.7));
fill((0,9/2*dayheight - 1/2*barheight)--(10*unitwidth,9/2*dayheight - 1/2*barheight)--(10*unitwidth,9/2*dayheight + 1/2*barheight)--(0,9/2*dayheight + 1/2*barheight)--cycle,gray(0.5));
draw((10*unitwidth,0)--(10*unitwidth,5*dayheight),gray(0.7));
fill((0,3/2*dayheight - 1/2*barheight)--(11*unitwidth,3/2*dayheight - 1/2*barheight)--(11*unitwidth,3/2*dayheight + 1/2*barheight)--(0,3/2*dayheight + 1/2*barheight)--cycle,gray(0.5));
draw((11*unitwidth,0)--(11*unitwidth,5*dayheight),gray(0.7));
draw((12*unitwidth,0)--(12*unitwidth,5*dayheight),gray(0.7));
fill((0,7/2*dayheight - 1/2*barheight)--(13*unitwidth,7/2*dayheight - 1/2*barheight)--(13*unitwidth,7/2*dayheight + 1/2*barheight)--(0,7/2*dayheight + 1/2*barheight)--cycle,gray(0.5));
draw((0*unitwidth,0)--(0*unitwidth,5*dayheight),gray(0.7));
draw((13*unitwidth,0)--(13*unitwidth,5*dayheight),gray(0.7));
draw((14*unitwidth,0)--(14*unitwidth,5*dayheight),gray(0.7));
label("$0$", (0,5*dayheight), N);
label("$4$", (2*unitwidth,5*dayheight), N);
label("$8$", (4*unitwidth,5*dayheight), N);
label("$12$", (6*unitwidth,5*dayheight), N);
label("$16$", (8*unitwidth,5*dayheight), N);
label("$20$", (10*unitwidth,5*dayheight), N);
label("$24$", (12*unitwidth,5*dayheight), N);
label("$28$", (14*unitwidth,5*dayheight), N);
label("Number of students at soccer practice", (7*unitwidth,6*dayheight));
label("Monday", (-0.5*unitwidth,9/2*dayheight), W);
label("Tuesday", (-0.5*unitwidth,7/2*dayheight), W);
label("Wednesday", (-0.5*unitwidth,5/2*dayheight), W);
label("Thursday", (-0.5*unitwidth,3/2*dayheight), W);
label("Friday", (-0.5*unitwidth,1/2*dayheight), W);
[/asy]
$\textbf{(A) } \text{The mean increases by 1 and the median does not change.}$
$\textbf{(B) } \text{The mean increases by 1 and the median increases by 1.}$
$\textbf{(C) } \text{The mean increases by 1 and the median increases by 5.}$
$\textbf{(D) } \text{The mean increases by 5 and the median increases by 1.}$
$\textbf{(E) } \text{The mean increases by 5 and the median increases by 5.}$
2010 AMC 8, 1
At Euclid High School, the mathematics teachers are Mrs. Germain, Mr. Newton, and Mrs. Young. There are $11$ students in Mrs. Germain's class, 8 in Mr. Newton, and $9$ in Mrs. Young's class are taking the AMC $8$ this year. How many mathematics students at Euclid High School are taking the contest?
$ \textbf{(A)}\ 26 \qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 28\qquad\textbf{(D)}\ 29\qquad\textbf{(E)}\ 30 $
2019 AMC 8, 2
Three identical rectangles are put together to form rectangle $ABCD$, as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles $5$ feet, what is the area in square feet of rectangle $ABCD$?
[asy]draw((0,0)--(0,10)--(15,10)--(15,0)--(0,0));
draw((0,5)--(10,5));
draw((10,0)--(10,10));
label("$A$",(0,0),SW);
label("$B$",(15,0),SE);
label("$C$",(15,10),NE);
label("$D$",(0,10),NW);
dot((0,10));
dot((15,0));
dot((15,10));
dot((0,0));
[/asy]
$\textbf{(A) }45\qquad
\textbf{(B) }75\qquad
\textbf{(C) }100\qquad
\textbf{(D) }125\qquad
\textbf{(E) }150\qquad$
2025 AMC 8, 19
Two towns, $A$ and $B$, are connected by a straight road, $15$ miles long. Traveling from town $A$ to town $B$, the speed limit changes every $5$ miles: from $25$ to $40$ to $20$ miles per hour (mph). Two cars, one at town $A$ and one at town $B$, start moving toward each other at the same time. They drive exactly the speed limit in each portion of the road. How far from town $A$, in miles, will the two cars meet?
$\textbf{(A) }7.75 \qquad\textbf{(B) }8 \qquad\textbf{(C) }8.25\qquad\textbf{(D) }8.5 \qquad\textbf{(E) }8.75$
2013 AMC 8, 7
Trey and his mom stopped at a railroad crossing to let a train pass. As the train began to pass, Trey counted 6 cars in the first 10 seconds. It took the train 2 minutes and 45 seconds to clear the crossing at a constant speed. Which of the following was the most likely number of cars in the train?
$\textbf{(A)}\ 60 \qquad \textbf{(B)}\ 80 \qquad \textbf{(C)}\ 100 \qquad \textbf{(D)}\ 120 \qquad \textbf{(E)}\ 140$
2025 AMC 8, 1
The eight pointed star is a popular quilting pattern. What percent of the entire 4-by-4 grid is covered by the star?
$(A)40$ $~~~$ $(B)50$ $~~~$ $(C)60$ $~~~$ $(D)75$ $~~~$ $(E)80$
2004 AMC 8, 18
Five friends compete in a dart-throwing contest. Each one has two darts to throw at the same circular target, and each individual's score is the sum of the scores in the target regions that are hit. The scores for the target regions are the whole numbers $1$ through $10$. Each throw hits the target in a region with a different value. The scores are: Alice $16$ points, Ben $4$ points, Cindy $7$ points, Dave $11$ points, and Ellen $17$ points. Who hits the region worth $6$ points?
$\textbf{(A)}\ \text{Alice} \qquad
\textbf{(B)}\ \text{Ben}\qquad
\textbf{(C)}\ \text{Cindy}\qquad
\textbf{(D)}\ \text{Dave}\qquad
\textbf{(E)}\ \text{Ellen}$
2016 AMC 8, 13
Two different numbers are randomly selected from the set ${ - 2, -1, 0, 3, 4, 5}$ and multiplied together. What is the probability that the product is $0$?
$\textbf{(A) }\dfrac{1}{6}\qquad\textbf{(B) }\dfrac{1}{5}\qquad\textbf{(C) }\dfrac{1}{4}\qquad\textbf{(D) }\dfrac{1}{3}\qquad \textbf{(E) }\dfrac{1}{2}$
2016 AMC 8, 5
The number $N$ is a two-digit number.
[list]
[*]When $N$ is divided by $9$, the remainder is $1$.
[*]When $N$ is divided by $10$, the remainder is $3$.
[/list]
What is the remainder when $N$ is divided by $11$?
$\textbf{(A) }0\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad \textbf{(E) }7$
2015 AMC 8, 3
Jack and Jill are going swimming at a pool that is one mile from their house. They leave home simultaneously. Jill rides her bicycle to the pool at a constant speed of $10$ miles per hour. Jack walks to the pool at a constant speed of $4$ miles per hour. How many minutes before Jack does Jill arrive?
$\textbf{(A) }5\qquad\textbf{(B) }6\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad \textbf{(E) }10$
2020 AMC 8 -, 20
A scientist walking through a forest recorded as integers the heights of $5$ trees standing in a row. She observed that each tree was either twice as tall or half as tall as the one to its right. Unfortunately some of her data was lost when rain fell on her notebook. Her notes are shown below, with blanks indicating the missing numbers. Based on her observations, the scientist was able to reconstruct the lost data. What was the average height of the trees, in meters?
$$
\begingroup
\setlength{\tabcolsep}{10pt}
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{|c|c|}
\hline Tree 1 & \rule{0.4cm}{0.15mm} meters \\
Tree 2 & 11 meters \\
Tree 3 & \rule{0.5cm}{0.15mm} meters \\
Tree 4 & \rule{0.5cm}{0.15mm} meters \\
Tree 5 & \rule{0.5cm}{0.15mm} meters \\ \hline
Average height & \rule{0.5cm}{0.15mm}\text{ .}2 meters \\
\hline
\end{tabular}
\endgroup$$
$\newline \textbf{(A) }22.2 \qquad \textbf{(B) }24.2 \qquad \textbf{(C) }33.2 \qquad \textbf{(D) }35.2 \qquad \textbf{(E) }37.2$
2024 AMC 8 -, 16
Minh enters the numbers from 1 to 81 in a $9\times9$ grid in some order. She calculates the product of the numbers in each row and column. What is the least number of rows and columns that could have a product divisible by 3?
$\textbf{(A) }8\qquad\textbf{(B) }9\qquad\textbf{(C) }10\qquad\textbf{(D) }11\qquad\textbf{(E) }12$
2019 AMC 8, 4
Quadrilateral $ABCD$ is a rhombus with perimeter $52$ meters. The length of diagonal $\overline{AC}$ is $24$ meters. What is the area in square meters of rhombus $ABCD$?
[asy]
unitsize(1cm);
draw((0,1)--(2,2)--(4,1)--(2,0)--cycle);
dot("$A$",(0,1),W);
dot("$D$",(2,2),N);
dot("$C$",(4,1),E);
dot("$B$",(2,0),S);
[/asy]
$\textbf{(A) } 60
\qquad\textbf{(B) } 90
\qquad\textbf{(C) } 105
\qquad\textbf{(D) } 120
\qquad\textbf{(E) } 144$